10m or 25m? Contextually important.
You using thunder focus tea on cd? Which healing talent did you take, are you effectively fistweaving (maintaining eminence? statue is good position?) Do you have competent raiders who will walk on healing spheres? what's your glyph setup? reforging?

What combination of bosses did you have for the first one (I'm assuming there's a possibility of each realm having a different one). We had a ranged group and a loose melee stack for ours with Jasper, Jade, and Amethyst so that melee wasn't always getting a full blitz of the pools while keeping ranged close enough so we never worried about the jasper chains (ranged moved in and stuck with their melee counterpart while staying no more than 10 yards away).

I found that I was doing just fine on mana the entire time. Even when our druid healer went down, it wasn't until just as we were wiping that my mana felt super strained. I'm also not glyphed for mana tea (I meant to but completely forgot till we were mid raid). I found that during Jade petrification the overall raid damage was so low I could take a minute to channel mana tea while the druid and pally were still keeping tanks up. A lot of that damage was avoidable. I'm also at about 7800 spirit unbuffed. Add in a 1k spirit flask and 600 spirit from food (I'm a panda) and I'm sitting at 9400 spirit. This may be the holy priest in me (previous main) but I stacked spirit like a champ and had no regrets about it.

My statue was placed about halfway from where the boss starts to the stairs leading back up from where you fight them so my soothing mist from it could reach about anybody. I also didn't so much "fistweave" as more used jab just as a guranteed chi generator because when I find that if I need chi soothing mist won't cut it and renewing mist has that whole cooldown thing going for it. My top heal for that fight was easily uplift, and I was using the Chi brew for more extreme burst in the case that a tranq wasn't cutting it or a raid cd didn't go off fast enough after an unmitigated overload.

Sounds like people were taking too much damage. If you're constantly having to burst heal people with Surging Mist, you'll definitely go OOM fast. The top culprit for us was people not getting close fast enough on Jasper Chains or standing in the Cobalt Mines (big blue crystals).

I was fistweaving on it with 5.5k spirit (including 1000 from flask--my gear sucked) and didn't run out of mana until closer to 20%, which was like 1 minute before enrage for us. I still did some of the top healing among the raid healers (not really fair to compare to the three tank healers, since obviously I was beating them in healing). Mana is probably a bit easier in 25s, just because we had a mana tide and could abuse the shadow priests for Hymn of Hope.

I think we had the exact same combo: Jade, Jasper, and Amethyst. At first, we tried to brute force through the Jasper Chains but it was way too much single target burst healing required at our ilvl. Then we move together as a single group. That made the healing requirement much easier. I was running at 3.7k spirit fistweaving (failing with no flask and ran out of spi food) and still come out of the fight with ~20% mana.

I also did some quick excel math to see what's the most efficient chi gen + chi spending rotation for a Mistweaver before the raid so that might have helped.

I think at that point (2 gear upgrades ago) I was at about 8.9k spirit unbuffed -- now at 9.3k. In any case, over 10k with flask+food. I was fistweaving and really not having many mana issues until the end of the fight. Same combo by the way. Actually, since we didn't really read the journal as thoroughly as we should have, we took an absolutely excessive amount of damage from chains. We killed it, but I was comparing our logs to another guild on our server that failed to kill it but had an attempt with a similar length to ours... we took an additional 20 million damage from chains over them since we weren't waiting for petrification. You're not making that mistake are you?